Abstract

We prove the existence of one-signed periodic solutions of second-order nonlinear difference equation on a finite discrete segment with periodic boundary conditions by combining some properties of Green's function with the fixed-point theorem in cones.

1. Introduction

Let be the set of real numbers, be the integers set, with , and .

In recent years, the existence and multiplicity of positive solutions of periodic boundary value problems for difference equations have been studied extensively, see [1–5] and the references therein. In 2003, Atici and Cabada [2] studied the existence of solutions of second-order difference equation boundary value problem
where , satisfy

(H1) ;

(H2) is continuous with respect to .

The authors obtained the existence results of solutions of (1.1) under conditions (H1), (H2), and the used tool is upper and lower solutions techniques.

Naturally, whether there exists the Green function of the homogeneous linear boundary value problem corresponding to (1.1) if ? Moreover, if the answer is positive, whether keeps its sign? To the knowledge of the authors, there are very few works on the case .

Recently, in 2003, Torres [6] investigated the existence of one-signed periodic solutions for second-order differential equation boundary value problem
by applying the fixed-point theorem in cones, and constructed Green’s function of
where satisfies either

(H3) , on ;

(H4) , on and for some .

Motivated by Torres [6], in Section 2, the paper gives the new expression of Green’s function of the linear boundary value problem
where and
and obtains the sign properties of Green’s function of (1.4), (1.5).

In Section 3, we obtain the existence of one-signed periodic solutions of the discrete second-order nonlinear periodic boundary value problem
where is continuous. For related results on the associated differential equations, see Torres [6].

2. Preliminaries

Let
be a Banach space endowed with the norm .

We say that the linear boundary value problem (1.4), (1.5) is nonresonant when its unique solution is the trivial one. If (1.4), (1.5) is nonresonant, and let , by the virtue of the Fredholm’s alternative theorem, we can get that the discrete second-order periodic boundary value problem
has a unique solution ,
where is Green’s function related to (1.4), (1.5).

Definition 2.1 (see [7]). We say that a solution of (1.4) has a generalized zero at provided that if and if either or .

Theorem 2.2. Assume that the distance between two consecutive generalized zeros of a nontrivial solution of (1.4) is greater than . Then Green’s function has constant sign.

Proof. Obviously, is well defined on . We only need to prove that has no generalized zero in any point. Suppose on the contrary that there exists such that is a generalized zero of . It is well known that for a given , as a function of is a solution of (1.4) in the intervals and such that
Case 1 (). If , we can construct
Consequently, is a solution of (1.4) in the whole interval . Since , we have , that is, . Moreover, , so there at least exists another generalized zero of . Note that the distance between and is smaller than , which is a contradiction.Analogously, if , we get a contradiction by the same reasoning with
If , we can apply as defined (2.7). Since and , which contradicts with the hypothesis.Case 2 (). If , we can construct defined as (2.6). It is not difficult to verify that is a solution of (1.4) in the whole interval . Also, we have that , that is, is a generalized zero of . Moreover, , so there at least exists another generalized zero of . Note that the distance between and is smaller than , which is a contradiction.Similarly, if , we can get a contradiction by the same reasoning as defined (2.7).If , we can construct defined by (2.7). Since and , it is clear that there exists another generalized zero of . Note that the distance between and is smaller than , this contradicts with the hypothesis.

To apply the above result, we are going to study the two following cases.

Corollary 2.3. If , then for all .

Proof. If , by [7, Corollary 6.7], it is easy to verify that (1.4) is disconjugate on , and any nontrivial solution of (1.4) has at most one generalized zero on . Hence, by Theorem 2.2, Green’s function has constant sign. We claim that the sign is negative. In fact, is the unique -periodic solution of the equation
and summing both sides of (2.8) from to , we can get
Since for some , and as a consequence for all .

Remark 2.4. If ( is a negative constant), then by computing we can obtain
where . Obviously, .

If , then the solutions of (1.4) are oscillating, that is, there are infinite zeros, and to get the required distance between generalized zeros, should satisfy .

Corollary 2.5. If , then for all .

Proof. We claim that the distance between two consecutive generalized zeros of a nontrivial solution of (1.4) is strictly greater than . In fact, it is not hard to verify that is disconjugate on under assumption . Since , by Sturm comparison theorem [7, Theorem 6.19], (1.4) is disconjugate on , that is, any nontrivial solution of (1.4) has at most one generalized zero on .Hence, by Theorem 2.2, has constant sign on , and the positive sign of is determined as the proof process of Corollary 2.3.

Remark 2.6. If , and , then by computing we can obtain
where and . Clearly, .If and , then , and by computing we get
Obviously, Green’s function for and for .If and , then , and it is not difficult to verify that
are nontrivial solutions of (1.4), (1.5). That is, the problem (1.4), (1.5) has no Green’s function.If , then Green’s function may change its sign. For example, let , it is easy to verify that and , thus
Clearly, for , for , and for .Consequently, is the optimal condition of .

Next, we provide a way to get the expression of . Let be the unique solution of the initial value problem
and be the unique solution of the initial value problem

Lemma 2.7. Let . Then Green’s function of (1.4), (1.5) is explicitly given by

Proof. Suppose that Green’s function of (1.4), (1.5) is of the form
where can be determined by imposing the boundary conditions.From the basis theory of Green’s function, we know that
Hence, , combining with , we can get
Moreover, since , it follows that

Note that has the same sign with . In fact, by the comparison theorem [7, Theorem 6.6], it is easy to prove that on . If , then
Thus . Similarly, we can get that . Since , we have
That is .

If , by the similar method, we can prove .

Lemma 2.8. Let . Then the periodic boundary value problem (2.2), (2.3) has the unique solution
where is defined by (2.17).

Proof. We check that satisfies (2.2). In fact,
On the other hand, it is easy to verify that .

Denote that
As a direct application, we can compute the maximum and the minimum of the Green’s function when , it follows that
where is defined in Remark 2.4. Similarly, when , we can get
where is defined in Remark 2.6.

3. Main Results

In this section, we consider the existence of one-signed solutions of (1.7). The following well-known fixed-point theorem in cones is crucial to our arguments.

Theorem 3.1 (see [8]). Let be a Banach space and be a cone. Suppose and are bounded open subsets of with . Assume that is a completely continuous operator such that either(i) and or(ii) and .
Then has a fixed point in .

Theorem 3.2. Assume that there exist and such that
If one of the following conditions holds:(i)(ii)then problem (1.7) has a positive solution.

Proof. From Corollary 2.5, we get that . It is easy to see that the equation is equivalent to
Define the open sets
and the cone in ,
Clearly, if , then .From Lemma 2.8, we define the operator by
From (3.1), if , then
Thus . Moreover, is a finite space, it is easy to prove that is a completely continuous operator. Clearly, is the solution of problem (1.7) if and only if is the fixed point of the operator .We only prove (i). (ii) can be obtained by the similar method. If , then and for all . Therefore, from (i),
If , then and for all . As a consequence,
From Theorem 3.1, has a fixed point and satisfies
Therefore, is a positive solution of (1.7).